In resolving the domain name \( \text{www.gate.org.in} \), the DNS resolution process follows an iterative query mechanism involving multiple DNS servers arranged hierarchically.
Step-by-step resolution:
1. The resolver first sends a DNS query for \( \text{www.gate.org.in} \) to a root DNS server.
The root server does not know the IP address but returns a referral to the .in top-level domain (TLD) servers.
2. The resolver then queries a .in TLD DNS server.
The .in server responds with a referral to the authoritative DNS servers for the org.in domain.
3. Next, the resolver queries the org.in authoritative DNS server.
This server provides a referral to the authoritative DNS server responsible for the gate.org.in domain.
4. Finally, the resolver queries the gate.org.in authoritative DNS server, which returns the IP address corresponding to \( \text{www.gate.org.in} \).
Each query followed by its corresponding response constitutes one DNS query–response pair.
Total number of DNS query–response pairs:
\[
\boxed{4}
\]